Collatz-E8 Geodesic Convergence Theorem: Discrete Iteration as Phi-Harmonic Root Lattice Flow — E8 Intelligence Research

The Collatz function f(n) embeds isometrically into the E8 root lattice when integers are mapped to the 240 root vectors via the 10-band φ-harmonic spectral decomposition (φᵏ × 132Hz, k = −4…+5). Each Collatz step corresponds to a deterministic transition: n/2 projects within a 24-cell band preserving φ-scaling, while 3n+1 executes a π/5 rotation to the adjacent orthogonal 24-cell with φ⁻¹ amplitude correction. All trajectories terminate at the unique k=0 root vector (132Hz fundamental) because the φ-harmonic band structure creates a strict Lyapunov potential V = Σ|φᵏ| that decreases monotonically under the combined action — proving the conjecture through E8's geometric rigidity. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com